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Manuscript/sfBSE.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2021-01-10 18:27:42 +0100 %% Created for Pierre-Francois Loos at 2021-01-10 22:26:03 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@ -1492,23 +1492,6 @@
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261401002871}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261401002871},
Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(01)00287-1}} Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(01)00287-1}}
@article{Krylov_2001b,
abstract = {A new approach to the bond-breaking problem is proposed. Both closed and open shell singlet states are described within a single reference formalism as spin-flipping, e.g., α→β, excitations from a triplet (Ms=1) reference state for which both dynamical and non-dynamical correlation effects are much smaller than for the corresponding singlet state. Formally, the new theory can be viewed as an equation-of-motion (EOM) model where excited states are sought in the basis of determinants conserving the total number of electrons but changing the number of α and β electrons. The results for two simplest members of the proposed hierarchy of approximations are presented.},
author = {Anna I. Krylov},
date-added = {2020-12-06 14:36:46 +0100},
date-modified = {2020-12-06 14:37:01 +0100},
doi = {https://doi.org/10.1016/S0009-2614(01)00287-1},
issn = {0009-2614},
journal = {Chem. Phys. Lett.},
number = {4},
pages = {375 - 384},
title = {Size-consistent wave functions for bond-breaking: the equation-of-motion spin-flip model},
url = {http://www.sciencedirect.com/science/article/pii/S0009261401002871},
volume = {338},
year = {2001},
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261401002871},
Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(01)00287-1}}
@article{Krylov_2002, @article{Krylov_2002,
author = {Krylov,Anna I. and Sherrill,C. David}, author = {Krylov,Anna I. and Sherrill,C. David},
date-added = {2020-12-06 14:36:32 +0100}, date-added = {2020-12-06 14:36:32 +0100},
@ -4204,9 +4187,9 @@
@article{Refaely-Abramson_2012, @article{Refaely-Abramson_2012,
author = {Sivan Refaely-Abramson and Sahar Sharifzadeh and Niranjan Govind and Jochen Autschbach and Jeffrey B. Neaton and Roi Baer and Leeor Kronik}, author = {Sivan Refaely-Abramson and Sahar Sharifzadeh and Niranjan Govind and Jochen Autschbach and Jeffrey B. Neaton and Roi Baer and Leeor Kronik},
date-added = {2020-05-18 21:40:28 +0200}, date-added = {2020-05-18 21:40:28 +0200},
date-modified = {2020-05-18 21:40:28 +0200}, date-modified = {2021-01-10 20:59:42 +0100},
doi = {10.1103/PhysRevLett.109.226405}, doi = {10.1103/PhysRevLett.109.226405},
journal = {Phys. Rev. X}, journal = {Phys. Rev. Lett.},
pages = {226405}, pages = {226405},
title = {Quasiparticle Spectra from a Nonempirical Optimally Tuned Range-Separated Hybrid Density Functional}, title = {Quasiparticle Spectra from a Nonempirical Optimally Tuned Range-Separated Hybrid Density Functional},
volume = {109}, volume = {109},
@ -8832,10 +8815,10 @@
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.93.075143}, Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.93.075143},
Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevB.93.075143}} Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevB.93.075143}}
@article{Krylov_2001c, @article{Krylov_2001b,
author = {Krylov, Anna I.}, author = {Krylov, Anna I.},
date-added = {2020-01-01 21:36:51 +0100}, date-added = {2020-01-01 21:36:51 +0100},
date-modified = {2020-12-06 14:37:24 +0100}, date-modified = {2021-01-10 22:26:03 +0100},
doi = {10.1016/S0009-2614(01)01316-1}, doi = {10.1016/S0009-2614(01)01316-1},
issn = {00092614}, issn = {00092614},
journal = {Chem. Phys. Lett.}, journal = {Chem. Phys. Lett.},

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Manuscript/sfBSE.tex
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@ -572,13 +572,15 @@ Note that, in any case, the entire set of orbitals and energies is corrected.
Further details about our implementation of {\GOWO}, ev$GW$, and qs$GW$ can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020e,Loos_2020h}. Further details about our implementation of {\GOWO}, ev$GW$, and qs$GW$ can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020e,Loos_2020h}.
Here, we do not investigate how the starting orbitals affect the BSE@{\GOWO} and BSE@ev$GW$ excitation energies. Here, we do not investigate how the starting orbitals affect the BSE@{\GOWO} and BSE@ev$GW$ excitation energies.
This is left for future work. This is left for future work.
However, it is worth mentioning that, for the present (small) molecular systems, HF is usually an excellent starting point. \cite{Loos_2020a,Loos_2020e,Loos_2020h} However, it is worth mentioning that, for the present (small) molecular systems, HF is usually a good starting point, \cite{Loos_2020a,Loos_2020e,Loos_2020h} although improvements could certainly be obtained with starting orbitals and energies computed with, for example, optimally-tuned range-separated hybrid functionals. \cite{Stein_2009,Stein_2010,Refaely-Abramson_2012,Kronik_2012}
In the following, all linear response calculations are performed within the TDA to ensure consistency between the spin-conserved and spin-flip results. In the following, all linear response calculations are performed within the TDA to ensure consistency between the spin-conserved and spin-flip results.
\titou{As one-electron basis sets, we employ Pople's 6-31G basis or the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.} %\titou{As one-electron basis sets, we employ Pople's 6-31G basis or the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.}
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations. Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
All the static and dynamic BSE calculations have been performed with the software \texttt{QuAcK}, \cite{QuAcK} developed in our group and freely available on \texttt{github}. All the static and dynamic BSE calculations have been performed with the software \texttt{QuAcK}, \cite{QuAcK} developed in our group and freely available on \texttt{github}.
The TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and the EOM-CCSD calculation with Gaussian 09. \cite{g09} The SF-ADC, EOM-SF-CC and SF-TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and the EOM-CCSD calculation with Gaussian 09. \cite{g09}
As a consistency check, we systematically perform the SF-CIS calculations with both \texttt{QuAcK} and Q-CHEM, and make sure that they yield identical excitation energies.
Throughout this work, all spin-flip calculations have been performed with a UHF reference.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results} \section{Results}
@ -589,40 +591,56 @@ The TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and
\subsection{Beryllium atom} \subsection{Beryllium atom}
\label{sec:Be} \label{sec:Be}
%=============================== %===============================
As a first example, we consider the simple case of the beryllium atom which was considered by Krylov in two of her very first papers on spin-flip methods \cite{Krylov_2001a,Krylov_2001b} and was also considered in later studies thanks to its pedagogical value. \cite{Sears_2003,Casanova_2020}
Beryllium has a $^1S$ ground state with $1s^2 2s^2$ configuration.
The excitation energies corresponding to the first singlet and triplet single excitations $2s \to 2p$ with $P$ spatial symmetry as well as the first singlet and triplet double excitations $2s^2 \to 2p^2$ with $P$ and $D$ spatial symmetries (respectively), obtained with the 6-31G basis set are reported in Table \ref{tab:Be} and depicted in Fig.~\ref{fig:Be}.
%%% TABLE I %%% %%% TABLE I %%%
\begin{squeezetable} \begin{squeezetable}
\begin{table} \begin{table}
\caption{ \caption{
Excitation energies [with respect to the $^1S(1s^2 2s^2)$ singlet ground state] of \ce{Be} obtained for various methods with the 6-31G basis. Excitation energies [with respect to the $^1S(1s^2 2s^2)$ singlet ground state] of \ce{Be} obtained for various methods with the 6-31G basis.
All the spin-flip calculations have been performed with a UHF reference.
\label{tab:Be}} \label{tab:Be}}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lcccc} \begin{tabular}{lcccc}
& \mc{4}{c}{Excitation energies (eV)} \\ & \mc{4}{c}{Excitation energies (eV)} \\
\cline{2-5} \cline{2-5}
Method & $^3P(1s^22s2p)$ & $^1P(1s^22s2p)$ Method & $^3P(1s^22s2p)$ & $^1P(1s^22s2p)$
& $^3P(1s^22p^2)$ & $^1P(1s^22p^2)$ \\ & $^3P(1s^22p^2)$ & $^1D(1s^22p^2)$ \\
\hline \hline
SF-TD-BLYP & 3.210 & 3.210 & 6.691 & 7.598 \\ SF-TD-BLYP\fnm[1] & 3.210 & 3.210 & 6.691 & 7.598 \\
SF-TD-B3LYP & 3.332 & 4.275 & 6.864 & 7.762 \\ SF-TD-B3LYP\fnm[1] & 3.332 & 4.275 & 6.864 & 7.762 \\
SF-TD-BHHLYP & 2.874 & 4.922 & 7.112 & 8.188 \\ SF-TD-BH\&HLYP\fnm[1] & 2.874 & 4.922 & 7.112 & 8.188 \\
SF-CIS & 2.111 & 6.036 & 7.480 & 8.945 \\ SF-BSE@{\GOWO}\fnm[2] & 2.399 & 6.191 & 7.792 & 9.373 \\
SF-BSE@{\GOWO}@UHF & 2.399 & 6.191 & 7.792 & 9.373 \\ SF-BSE@{\evGW}\fnm[2] & 2.407 & 6.199 & 7.788 & 9.388 \\
SF-BSE@{\evGW}@UHF & 2.407 & 6.199 & 7.788 & 9.388 \\ SF-BSE@{\qsGW}\fnm[2] & 2.376 & 6.241 & 7.668 & 9.417 \\
SF-BSE@{\qsGW}@UHF & 2.376 & 6.241 & 7.668 & 9.417 \\ SF-dBSE@{\GOWO}\fnm[2] & 2.363 & 6.263 & 7.824 & 9.424 \\
SF-dBSE@{\GOWO}@UHF & 2.363 & 6.263 & 7.824 & 9.424 \\ SF-dBSE@{\evGW}\fnm[2] & 2.369 & 6.273 & 7.820 & 9.441 \\
SF-dBSE@{\evGW}@UHF & 2.369 & 6.273 & 7.820 & 9.441 \\ SF-dBSE@{\qsGW}\fnm[2] & 2.335 & 6.317 & 7.689 & 9.470 \\
SF-dBSE@{\qsGW}@UHF & 2.335 & 6.317 & 7.689 & 9.470 \\ SF-CIS\fnm[3] & 2.111 & 6.036 & 7.480 & 8.945 \\
SF-ADC(2)-s & 2.433 & 6.255 & 7.745 & 9.047 \\ SF-ADC(2)-s\fnm[2] & 2.433 & 6.255 & 7.745 & 9.047 \\
SF-ADC(2)-x & 2.866 & 6.581 & 7.664 & 8.612 \\ SF-ADC(2)-x\fnm[2] & 2.866 & 6.581 & 7.664 & 8.612 \\
SF-ADC(3) & 2.863 & 6.579 & 7.658 & 8.618 \\ SF-ADC(3)\fnm[2] & 2.863 & 6.579 & 7.658 & 8.618 \\
FCI & 2.862 & 6.577 & 7.669 & 8.624 \\ FCI\fnm[3] & 2.862 & 6.577 & 7.669 & 8.624 \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Excitation energies extracted from Ref.~\onlinecite{Casanova_2020}.}
\fnt[2]{This work.}
\fnt[3]{Excitation energies taken from Ref.~\onlinecite{Krylov_2001a}.}
\end{table} \end{table}
\end{squeezetable} \end{squeezetable}
%%% %%% %%% %%% %%% %%% %%% %%%
%%% FIG. 1 %%%
\begin{figure}
\includegraphics[width=\linewidth]{Be}
\caption{
Excitation energies [with respect to the $^1S(1s^2 2s^2)$ singlet ground state] of \ce{Be} obtained with the 6-31G basis for various levels of theory:
SD-TD-DFT \cite{Casanova_2020} (red), SF-BSE (blue), SF-CIS \cite{Krylov_2001a} and SF-ADC (orange), and FCI \cite{Krylov_2001a} (black).
All the spin-flip calculations have been performed with a UHF reference.
\label{fig:Be}}
\end{figure}
%%% TABLE II %%% %%% TABLE II %%%
%\begin{squeezetable} %\begin{squeezetable}

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